Orientable:

The non-orientable surfaces form two classes, those based on the
real projective plane, which have odd
Euler characteristic, and those based on
the
Klein bottle,
which have even Euler characteristic. The surfaces in each class are
generated by adding handles to the base surface.

The only surfaces with odd Euler characteristic are non-orientable.

A non-orientable surface can not be
embedded
in three-space, but it can be
immersed there.